LRFD calibration of simple limit state functions in geotechnical soil-structure design - Risk and Reliability in Geotechnical Engineering - page 334

μ = Mean of g
σ = Standard deviation of g
β = Reliability index
P f = Probability of failure

μ

βσ

Failure
region
P f

γ Q
φ

0

g = X R - X Q

Figure 8.2 Distribution of factored limit state function g.

Or, equivalently by redefining g/Q n as g

γ
ϕ

Q

g = XX − (8.8)

R Q

The mean values of resistance and load bias values are denoted as μ R and μ Q , respectively,
and the corresponding coefficient of variation (COV) values as COV R and COV Q .
LRFD calibration can now be understood to be the selection of resistance and load factors
such that the probability that g is less than zero (Pf f (g < 0)) in Equation 8.8 does not exceed
a prescribed value. The distribution of g values for candidate values of γ Q and φ and random
variables X R and X Q is illustrated in Figure 8.2 . P f (g < 0) is related to the highlighted area
of the frequency distribution curve with g < 0.
Figure 8.2 also shows the definition of reliability index (β) as the number of standard
deviations (σ) between the mean (μ) of the distribution and g = 0. The relationship between
probability of failure and reliability index is

P f = 1 − Φ(β) (8.9)

Here, Φ is the standard normal cumulative distribution function (CDF) (NORM.DIST
in Excel). Common practice is to use the reliability index β to quantify margins of safety.
Equation 8.6 shows that the conventional meaning of factor of safety appears as F = γ Q /φ.
This equivalency provides a method to relate past practice using ASD to LRFD calibration
outcomes as discussed later in the chapter.

8.3 bIaS Value DIStrIbutIonS

The distributions for load and resistance bias values are best described using standard nor-
mal CDF plots in which the horizontal axis is bias X and the vertical axis is the standard
normal variable (z) computed as

z i = Φ −1 (p i ) (8.10)

Here, p i = i/(n + 1), n is the number of bias values and i = rank(Xi). i ). Normal distributions
of random variable X will plot as a straight line with μ X (mean of bias values) intersecting